Title Symbolic/Numeric genus computation. |
Version
française |

**Location**

INRIA, Project
CAFÉ

BP 93, 06902 France

**Information**

**Description**

The genus of an algebraic curve is a fundamental quantity that several algorithms in cryptography or computer algebra depend on. Computing it, which is often a preliminary step for other algorithms, remains difficult for general curves. In the past years, we have experimented with methods based on linear differential equations [2] and systems [1] associated with the curve. Using Hurwitz' formula, computing the genus is now reduced to computing matrices of residues at the roots of the discriminant of the curve, and then their eigenvalues. The goal of this internship is to replace the exact computation of those residue matrices and eigenvalues by their approximations using standard numerical algorithms, and then to recover the exact ramification indices from those numerical values. An error analysis should also be carried out in order to identify cases where the genus calculated in that way can be proven correct.

[1] M.Bronstein (1998): *
The Lazy Hermite Reduction,* INRIA Research Report 3562.

[2] O.Cormier, M.Singer, B.Trager and F.Ulmer (2002): *Linear
Differential Operators for Polynomial Equations,*
Journal of Symbolic Computation **34**, 5.

**Tools**

Unix workstation, the Maple computer algebra system, a numerical linear algebra library and the Aldor programming language.

**Duration**

3 months.